This is an investigation to find a relationship between the T-totals and the T-number. The diagram shows a 9x9 grid, with each individual cell having one number in it starting on the

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Introduction

Lianne Haley

COURSEWORK INVESTIGATION

T-Totals

The Problem, the Plan and possible extensions

This is an investigation to find a relationship between the T-totals and the T-number.

The diagram shows a 9x9 grid, with each individual cell having one number in it starting on the top row 1-9.

The diagram shown has an upright T-shape, the total of the numbers inside the T-shape is 1+2+3+11+20 = 37, and this is called the T-total. The number at the bottom of the T-shape is called the T-number. The T-number, for the example T-shape given, is 20.

I need to be systematic in my approach so initially I will be investigating the relationship between the T-totals and T-numbers when the T-shape translates on the 9x9 grid, starting with the 1st row then the 2nd row etc. This will keep it simple for me to spot any patterns.

Later I will be investigating T-shapes on different sized grids, again translating the T-shape to different positions on the grids to find a relationship between the T-totals and the T-numbers.

I can also use grids of different sizes again and try other transformations and combinations of transformations and investigate relationships between the T-totals, the T-numbers, the grid size and the transformations.

With a 90°clockwise transformation of an upright T the T-Number is neither the lowest nor the highest, but the highest number minus the T-Number is equal to 7.

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With a 270° clockwise transformation, the T-Number is, again, neither the highest nor the lowest number but the T-Number minus the lowest number is equal to 7.

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Justifying the Formula

Question

I need to find out the T-Total for a 270° T-Shape who’s T-Number is 55 on an 11x11 grid.

Answer

n=45

t=5n+7

t= (5x45) +7

t=225+7

t=232

Check

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Conclusion

I have found that each different translation requires a formula similar to each other translation for different sized grids.

I believe I have fully justified the explanation of this occurrence by checking my findings and tabulating results.

My initial hypothesis was correct stating that each T-Shape would increase by 5 thus finding the relationship between the T-Number and T-Total was +5 for any translation on any sized grid.

Further Extensions

If I had time I could explore the relationship between different sized T-Shapes on different sized grids, for example, extended T-Shapes (5 on the top row and 3 on the bottom row etc.) and elongated T-shapes.

We will now translate the L-shape into algebra bearing in mind that the L stands for L-number. L L+6 L+7 L+8 L+9 Here we get the following algebra: L + L + 9 + L + 8 + L + 7 + L + 6 When this algebraic equation is

5x101), with the two variables of grid width (g) and the T-Number (x). Finding relationship between T-Number and T-Total, with different sized grid and different translation, enlargements and rotations of the T-Shape For this section we will have to keep some items constant, as for to provide a stable environment to prove of disprove theories based on translations, enlargements